Install pip and packages from within Blender OS independently
Not satisfied with this, but a somehow hack-ish way:

import subprocess
import sys
from pathlib import Path  # Object-oriented filesystem paths since Python 3.4
# sys.prefix points on Windows to: 'D:\\*blender path*\\2.80\\python'
# Windows only: subprocess.call([str(Path(sys.prefix) / "bin" / "python.exe"), "-m", "ensurepip"])

# OS independent (Windows: bin\python.exe; Linux: bin/python3.7m)
py_path = Path(sys.prefix) / "bin"
py_exec = next(py_path.glob("python*"))  # first file that starts with "python" in "bin" dir
subprocess.call([str(py_exec), "-m", "ensurepip"])


You should see a blue 0 in your console in Blender.

Edit

If subprocess.call([str(py_exec), "-m", "ensurepip"]) results in a Permission Denied error (console shows 1):


Windows: Close Blender, right-click blender.exe and select Run as administrator. Then run the code again.
Linux (untested): Start Blender with the sudo command and run the code again.


Edit Blender 2.81:

Pip is now enabled by default from 2.81, so no need for this solution anymore: https://developer.blender.org/T71856

Installing a library in Blender with pip now works like this:

import subprocess
import ensurepip
ensurepip.bootstrap()
pybin = bpy.app.binary_path_python
subprocess.check_call([pybin, '-m', 'pip', 'install', 'your_package'])


Edit 2: Added pybin, '-m', otherwise it can install the package in a Python environment which is not Blender's one.

Edit 3: You first have to link pip with Blender (added ensurepip)